Thus, \(\mathbf {q}_A \in \mathcal {C}_{\alpha } \)  and \(\mathcal {C}_{\alpha } \)  is closed and uniformly integrable. In particular, for any \(p \in \mathbb {R}^{d}\) , \(\mathbf {q_A} \rightarrow \mathcal {L}(\mathbf {q_A},p) \)  is lower semicontinuous on \(\mathcal {P}_1(A) \) . Let \(\mathcal {C} = \left\lbrace  \mathbf {q}_A \in \mathcal {P}_1(A) , \int _A f(a) \mathbf {q}_A(da) \le H^*(q)+1 \right\rbrace  \) . Let us show that \(\mathcal {C}\)  is non-empty compact subset of \(\mathcal {P}_1(A)\) . \(\mathcal {C}\)  is not empty since it contains \(\delta _{a_0}\)  where \(a_0\)  is a point where \(f\)  (lower semi-continuous and coercive) achieves its minimum. From the coercivity assumption on \(f\)  we deduce that \(\mathcal {C}\)  is tight. But \(\mathcal {C} = \mathcal {C}_{\alpha } \)  for \(p=0\)  and \(\alpha = H^*(q)+1 \)  and what we did above yields uniform integrability and closedness for \(\mathcal {C}\) . Using [1]} Prop. 7.1.5, we can conclude that \(\mathcal {C}\)  is compact. We are then able to use the Von Neumann Theorem REF  to conclude that:
\( H^*(q) = \min _{\mathbf {q}_A \in \mathcal {P}(A)} \left\lbrace  \sup _{p\in \mathbb {R}^d} \left\lbrace  p.q + \int _A \left( f(a) - b(a).p \right) \mathbf {q}_A(da) \right\rbrace  \right\rbrace . \) 
